Henri Poincaré

Is the solar system stable forever? The 3-Body Problem A King and a Birthday Prize A discovered error and the discovery of a new field The N-Body Problem: http://members.fortunecity.com/kokhuitan/nbody.html Chaos and Henri Poincaré: http://zebu.uoregon.edu/~js/21st_century_science/readings/Parker_Chap3.html Try the 3 body problem! (read the instructions first): http://astro.u-strasbg.fr/~koppen/body/ThreeBody.html

Notes

Henri Poincaré, a mathematical physicist, attempted to answer the question of whether the solar system was stable forever, or if some planets would just simply drift off. This required an attempt to solve the celestial 3-body problem.

The 3 Body problem: Given 3 bodies (e.g. Sun, moon, Earth) and their initial positions and velocities, the problem is to determine the motion of the 3 bodies attracting one another according to Newtons law of gravity. Whilst the it sounds quite straightforward, the problem is surprisingly difficult to solve.

Issac Newton had solved the 2-Body problem and a solution was sought for the 3-Body problem and more generally the N-Body problem).

Given the deterministic way of thought, people believed that they could predict into the future provided they have sufficient information. Thus, given sufficient information they could easily solve the 3-Body problem.

In 1887 the King of Sweden and Norway, Oscar II, initiated a mathematical competition to celebrate his 60th Birthday in 1889. Henri Poincaré selected the 3-Body problem (actually, he considered a 9-Body problem: the then known about 8 planets plus the Sun. However, he realised that the minor components of the solar system would produce perturbations on the planets and thus the problem was closer to a 50-Body problem. He immediately saw the difficulty with this and restricted himself to the 3-Body problem.)

Poincaré was familiar with the then current algebraic techniques and their limitations. However, he started to look at the problem from a different point of view and decided to try a geometric approach. This approach was ground-breaking and although he had failed in solving the problem, he was awarded the prize.

His revolutionary work was to be published in Acta Mathematica. However, during the publication process, Edvard Phragmen (a Swedish Mathematician) noticed a serious error in Poincarés work. The editor of Acta Mathematica immediately stopped publication and asked Poincaré to review his work. With further effort, Poiincaré looked again at his data (primarily patterns on the slices in phase space) and realised that the orbit of a planet in a case such as his could not be calculated far into the future. He was shocked by the results and rewrote his paper.

After the final publication, Poincaré abandoned the 3-Body problem, although the strange results he had obtained bothered him and he made constant referrals to them throughout his lifetime. Given the lack of interest from the scientific community and the advent of computers to analyse efficiency the work, the 3-Body problem and it’s bizarre implications went out of vogue. It wasn’t until much later that the scientific community realised that Poincaré had predicted chaotic motion and broke the ground in the new field of chaos.